We proceed similarly as in [1]} and start with the integral representation
\(I_x(a,b)=\frac{1}{B(a,b)}\int _\xi ^{+\infty } t^{b-1}\mathrm {e}^{-at}\left(\frac{1-\mathrm {e}^{-t}}{t}\right)^{b-1}\, \mathrm {d}t.\) 
